**T.R. **

**SİİRT UNIVERSİTY **

** GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCE **

**SOME SPECIAL INTEGER SEQUENCES RELATED TO **

**BIPARTITE GRAPHS **

**M.Sc. THESIS **

**DIYAR OMAR MUSTAFA ZANGANA **

**(153114021) **

**Department of Mathematics **

**Supervisor: Asst. Prof. Dr. Abdulkadir KARAKAŞ **

**Co-Supervisor: Res. Asst. Dr. Ahmet ÖTELEŞ **

**August-2017 **

**SİİRT **

**ACCEPTANCE AND APPROVAL OF THESIS **

## The thesis entitled “SOME SPECIAL INTEGER SEQUENCES RELATED TO

## BIPARTITE GRAPHS” submitted by Diyar Omar Mustafa Zangana has been approved

## as Master Thesis at Department of Mathematics, Graduate School of Natural and Applied

## Sciences, Siirt University with unanimity/majority of votes by the committee listed below

## on …/…/…….

**Committee Members **

** Signature **

**Head **

## ………..

**Supervisor **

## ………..

**Member **

## ………..

## I approve the above results.

## Assoc. Prof. Dr. Koray ÖZRENK

## Director of the Graduate School of Natural and Applied Sciences

**ACKNOWLEDGMENTS **

## Firstly the name of Allah we started and starting.

## Before of all things, I want to thank God for helping me to finish my thesis, this

## is a great opportunity to acknowledge and to thanks all persons without their support, and

## help this thesis would have been impossible.

## I would like to add a few heartfelt words for the people who were part of this thesis in

## numerous ways.

## I wish to thanks my supervisors Arş.Gör.Dr.Ahmet Öteleş and

## Asst.Prof.Dr.Abdulkadir Karakaş for our indefatigable guidance, valuable suggestion,

## moral support, constant encouragement and contribution of time for the successful

## completion of thesis work. I am very grateful to there, for providing all the facilities

## needed during the thesis development.

## My special thanks to my family for indispensable support and encouragement

## thing my study. Finally, I would like to thank all those who helped me directly or

## indirectly.

## Diyar Omar Mustafa ZANGANA

## SİİRT-2017

**TABLE OF CONTENTS **

**Page **

**ACKNOWLEDGMENTS ... iii**

**TABLE OF CONTENTS ... iv**

**LIST OF TABLES ... vi**

**LIST OF FIGURES ... vii**

**LIST OF SYMBOLS ... viii**

**ÖZET ... ix**

**ABSTRACT ... x**

**1. INTRODUCTION ... 1**

## 1.1. Structure of the Thesis ... 2

**2. LITERATURE RESEARCH ... 3**

**3. PRELIMINARIES ... 14**

## 3.1. Special Types of Matrices ... 14

## 3.1.1. Diagonal matrices ... 14

## 3.1.2. Triangular matrices ... 14

## 3.1.3. Permutation matrices ... 14

## 3.1.4. Circulant matrices ... 15

## 3.1.5. Toeplitz matrices ... 15

## 3.1.6. Hankel matrices ... 16

## 3.1.7. Hessenberg matrices ... 16

## 3.1.8. Tridiagonal matrices ... 17

## 3.2. Graph Theory ... 17

## 3.3. Graphs and Matrices ... 22

## 3.3.1. Incidence Matrix ... 22

## 3.3.2. Adjacency Matrix ... 23

## 3.3.3. Laplacian Matrix ... 24

## 3.4. Bipartite Graph and Perfect Matching ... 25

## 3.5. Permanents ... 27

## 3.6. Number Sequences ... 29

**4. SOME SPECIAL INTEGER SEQUENCES RELATED TO BIPARTITE **

**GRAPHS ... 36**

## 4.1. Jacobsthal Numbers and Associated Bipartite Graphs ... 36

## 4.1.1. Maple procedure ... 40

## 4.2. Bipartite Graphs Associated with Circulant Matrices ... 41

**5. CONCLUSION ... 50**

**6. REFERENCES ... 51**

**LIST OF TABLES **

**Page **

**LIST OF FIGURES **

**Page **

**Figure 3.1. A graph with 6-vertices and 5-edges ... 18 **

**Figure 3.2. A graph with 5-vertices and 7-edges ... 18 **

**Figure 3.3. A graph with 5-vertices and 8-edges is called a (5; 8) graph ... 19 **

**Figure 3.4. Directed graph ... 19 **

**Figure 3.5. Undirected graph ... 20 **

**Figure 3.6. Directed multiple graph ... 20 **

**Figure 3.7. Un-directed multiple graph ... 20 **

**Figure 3.8. Un-directed Pseudo graph ... 21 **

**Figure 3.9. Directed Pseudo graph ... 21 **

**Figure 3.10. A graph with 6-vertices and 5-edges ... 21 **

**Figure 3.11. A graph with 5-vertices and 8-edges ... 22 **

**LIST OF SYMBOLS **

**Symbol **

**Explanation **

**𝑮 **

**: Graph **

## 𝑨

_{𝒏}

## = (𝒂

_{𝒊𝒋}

**) **

**: Matrix **

## 𝐴

_{𝑛}

## of order 𝑛 with entries 𝑎

_{𝑖𝑗}

## 𝑨

_{(𝒏,𝒌)}

## = (𝒂

_{𝒊𝒋}

**) **

**: Matrix **

## 𝐴

_{(𝑛,𝑘)}

## of order 𝑛 with entries 𝑎

_{𝑖𝑗}

## 𝑎

𝑖𝑗## = {

## 1, if − 1 ≤ |𝑗 − 𝑖| ≤ 𝑘 − 1

_{0, otherwise}

## 𝑮(𝑨

_{𝒏}

**) **

**: Graph with adjacency matrix **

## 𝐴

_{𝑛}

**𝑱(𝒏) **

**: **

## 𝑛th Jacobsthal number

**𝑭(𝒏) **

**: **

## 𝑛th Fibonacci number

**𝑳(𝒏) **

**: **

## 𝑛th Lucas number

## 𝑨

_{𝒏}𝒌

_{ }

_{ }

_{: }

_{: }

_{𝑘th contraction of the matrix 𝐴}

𝑛
## 𝐩𝐞𝐫𝑨

_{𝒏}

**: Permanent of the matrix **

## 𝐴

_{𝑛}

**ÖZET **

**YÜKSEK LİSANS **

**İKİ PARÇALI GRAFLARLA İLİŞKİLİ BAZI ÖZEL TAMSAYI DİZİLERİ **

**Diyar Omar Mustafa ZANGANA **

**Siirt Üniversitesi Fen Bilimleri Enstitüsü **

**Matematik Anabilim Dalı **

**Danışman **

**: Yrd. Doç. Dr. Abdulkadir KARAKAŞ **

**II. Danışman : Arş. Gör. Dr. Ahmet ÖTELEŞ **

**2017, 53 Sayfa **

Bu tezde, iki parçalı komşuluk matrisleri 𝑛 × 𝑛 (0,1) − matrisler olan bazı belli tip iki parçalı grafları ele aldık. Sonra bu iki parçalı grafların mükemmel eşlemeleri sayılarının çok iyi bilinen sayı dizilerine (örneğin; Fibonacci Lucas Jacobsthal) eşit olduğunu gösterdik. Daha sonra, ele aldığımız bu graflar ve onların mükemmel eşlemeleriyle ilgili bazı örnekler verdik. Son olarak, bu iki parçalı grafların mükemmel eşlemeleri sayılarını hesaplamak için Maple 2016 prosedürleri sunduk.

**Anahtar Kelimeler: Fibonacci dizisi, İki parçalı graf, Jacobsthal dizisi, Lucas dizisi, Mükemmel **
eşleme, Permanent.

## x

**ABSTRACT **

**M.Sc. THESIS **

**SOME SPECIAL INTEGER SEQUENCES RELATED TO **

**BIPARTITE GRAPHS **

**Diyar Omar Mustafa ZANGANA **

**The Graduate School of Natural and Applied Science of Siirt University **

**The Degree of Master of Science **

** In Mathematics **

**Supervisior **

**: Asst. Prof. Dr. Abdulkadir KARAKAŞ **

**Co-Supervisior : Res. Asst. Dr. Ahmet ÖTELEŞ **

**2017, 53 Pages **

In this thesis, we consider some certain types of bipartite graphs whose bipartite adjacency matrices are the 𝑛 × 𝑛 (0,1) − matrices. Then we show that the numbers of perfect matchings (1 − factosrs) of these bipartite graphs are equal to the famous integer sequences (e.g. Fibonacci, Lucas, Jacobsthal). After that, we give some examples concerned with these graphs and their perfect matchings. Finally, we present Maple procedures in order to calculate the numbers of perfect matchings of the bipartite graphs.

**Keywords: Bipartite graph, Fibonacci sequence, Jacobsthal sequence, Lucas sequence, Perfect **
matching, Permanent.

## 1. INTRODUCTION

## Bipartite graph is a graph in which the vertices can be divided into two parts

## such that no two vertices in the same part are joined by an edge. The investigation of

## the properties of bipartite graphs was begun by König. His work was motivated by

## an attempt to give a new approach to the investigation of matrices on determinants

## of matrices. As a practical matter, bipartite graphs form a model of the interaction

## between two di¤erent types of objects. For example; social network analysis, railway

## optimization problem, marriage problem etc (Asratian et al., 1998).

## A perfect matching (1-factors) of a graph is a matching (i.e., an independent

## edge set) in which every vertex of the graph is incident to exactly one edge of

## the matching. "The enumeration or actual construction of perfect matching of a

## bipartite graph has many applications, for example, in maximal ‡ow problems and

## in assignment and scheduling problems arising in operational research" (Minc, 1978).

## The number of perfect matchings of bipartite graphs also plays a signi…cant role in

## organic chemistry (Wheland, 1953).

## Fibonacci, Lucas and Jacobsthal numbers which are respectively de…ned by

## the recurrence relation

## F (n) = F (n

## 1) + F (n

## 2) ;

## F (0) = 0

## and F (1) = 1

## L (n) = L (n

## 1) + L (n

## 2) ;

## L (0) = 2

## and L (1) = 1

## J (n) = J (n

## 1) + 2J (n

## 2) ;

## J (0) = 0

## and J (1) = 1,

## for n

## 2, belong to a large family of positive integers. They have many interesting

## properties and applications to almost every …eld of science and art. They continue

## to contribute signi…cant innovations for investigations, and reveal the niceness of

## mathematics in many …elds, particularly number theory (Koshy, 2001; Koshy, 2011;

## Horadam, 1988).

## "The permanent of an n

## n

## matrix A = (a

ij## )

## is de…ned by

## per (A) =

## X

Sn n## Y

i=1## a

i (i)## where the summation extends over all permutations

## of the symmetric group S

n## .

## The permanent of a matrix is analogous to the determinant, where all of the signs

## used in the Laplace expansion of minors are positive" (Minc, 1978). Permanents have

## many applications in physics, chemistry, electrical engineering, graph theory etc.

## Some of the most important applications of permanents are via graph theory. A more

## di¢ cult problem with many applications is the enumeration of perfect matchings of

## has been very popular problem.

## 1.1. Structure of the Thesis

## The rest of the thesis is structured as follows.

## In Chapter 2, we present a discussion of previously published work that I

## did in this area in conjunction with other authors.

## In Chapter 3, we give the fundamental de…nitions, structures and theorems

## which are necessary to better understand the topics contained within this text.

## In Chapter 4, we consider a bipartite graph. Then we show that the

## num-bers of perfect matchings of this graph generate the Jacobsthal numnum-bers by the

## contraction method. Finally, a Maple procedure is presented in order to compute

## the numbers of perfect matchings of the graph.

## In Chapter 5, we …rstly introduce two lemmas ralated to bipartite graphs

## as-sociated with Fibonacci numbers. After that, we de…ne a bipartite graph asas-sociated

## with n

## n (0; 1)-circulant matrix whose the numbers of perfect matchings generate

## the Lucas numbers. Finally, two Maple procedures are presented to compute the

## numbers of perfect matchings in these graphs.

## 2.

## LITERATURE RESEARCH

## The purpose of this chapter is to further motivate the rest of the thesis

## by presenting a discussion of previously published work that I did in this area in

## conjunction with other authors.

## Lee et al. (1997) consider "a bipartite graph G F

(n;2)## = (f

i;j## )

## with bipartite

## adjacency matrix is the n

## n

## tridiagonal matrix of the form

## F

(n;2)## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 1

## 0

## 0

## 1

## 1

## 1

## . ..

## ..

## .

## 0

## 1

## . .. ... ... ...

## ..

## .

## . .. ... ... 1 0

## ..

## .

## . .. 1

## 1

## 1

## 0

## 0

## 1

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## ;

## (1)

## with the entries are

## f

ij## =

## (

## 1;

_{if jj}

## i

_{j}

## 1;

## 0;

## otherwise

## ". Then they obtain "the number of perfect matchings of G F

(n;2)## is the (n + 1)st

## Fibonacci number F (n + 1)". In other words,

## per

_{F}

(n;2) ## = F (n + 1) :

## (2)

## They also consider "a bipartite graph G F

(n;k)## = (f

i;j## )

## with bipartite adjacency

## matrix is the n

## n (0; 1)-matrix of the form

## F

(n;k)## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 1

## 1

## 0

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## . .. ... ... ...

## . .. ... ...

## ..

## .

## . ..

## . .. 0

## ..

## .

## . ..

## 1

## . .. ... ... ...

## 0

## 0

## 1

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## (3)

## with the entries are

## f

ij## =

## (

## 1;

## if

## 1

_{jj}

## i

_{j}

## k

## 1;

## ". Then they obtain "the number of perfect matchings of G F

(n;k)## is g

k## (n + k

## 1) ;

## where

## g

k_{(n)}

_{is the nth k-Fibonacci number" (Lee et al., 1997). This time, they consider}

## "an another bipartite graph G (U

n## = (u

i;j## ))

## with bipartite adjacency matrix is the

## n

## n (0; 1)-matrix of the form

## U

n## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 1

## 1

## 1

## 0

## 1

## 1

## 0

## 1

## 0

## 1

## 1

## ..

## .

## . .. ... ... ... ...

## ..

## .

## . .. 1

## 0

## 1

## 0

## 0

## 1

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## with the entries are

## u

ij## =

## 8

## >

## <

## >

## :

## 1;

## if i = j = 1 or i = j = n;

## 1;

## if i < j or j

## i =

## 1

## 0;

## otherwise

## ". Then they obtain "the number of perfect matchings of G (U

n## )

## is the (n + 1)st

## Fibonacci number F (n + 1)" (Lee et al., 1997).

## Lee (2000) considers "a bipartite graph G L

(n;2)## = (l

i;j## )

## with bipartite

## adjacency matrix is the n

## n

## matrix of the form

## L

(n;2)## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 0

## 1

## 0

## 0

## 1

## 1

## 1

## 0

## ..

## .

## 0

## 1

## . .. ... ... 0

## ..

## .

## . .. ... ... ... 0

## ..

## .

## . .. ... ... 1

## 0

## 0

## 1

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## ;

## with the entries are

## l

i;j## =

## 8

## >

## >

## <

## >

## >

## :

## 1;

## if i = j = 1 or i = 1 and j = 3;

## 1;

_{if i > 2 and jj}

## i

_{j}

## 1;

## 0;

## otherwise

## ". Then he shows that "the number of perfect matchings of G L

(n;2)## is (n

## 1)st

## Lucas number L (n

## 1)

## for n

## 3

_{". He also considers "a bipartite graph G L}

(n;k)
## with bipartite adjacency matrixL

(n;k)## =

## F

(n;k)## + E

1;k+1## P

k_{j=2}

## E

1;j## for n

## 3, where

## j)-entry and zeros elsewhere". Namely,

## L

(n;k)## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 0

## 0

## 1

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## . .. ... ... ...

## . .. ... ...

## ..

## .

## . ..

## . .. 0

## ..

## .

## . ..

## 1

## . .. ... ... ...

## 0

## 0

## 1

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## Then he shows that "the number of perfect matchings of G L

(n;k)## is l

k## (n

## 1) ;

## where

## l

k_{(n)}

_{is the nth k-Lucas number" (Lee, 2000). This time, he de…nes "the matrix}

## B

n## as

## B

n## =

## F

(n;2)## + E

13## E

23## + E

24## E

34## where F

(n;2)## is the matrix in (1) and E

i;j## denotes the n

## n

## matrix with 1 at the

## (i; j)-entry and zeros elsewhere". Let G (B

n## )

## be the bipartite graph with bipartite

## adjacency matrix B

n## :

## Then he shows that "the number of perfect matchings of

## G (B

n## )

## is (n

## 1)st Lucas number L (n

## 1)" (Lee, 2000).

## Shiu et al. (2003) …rstly de…ne the (k; )-sequences s

k_{(n) :}

_{Then they give}

## the following result:

## "For a …xed m

## 1, suppose n; k

## 2

## and n

## m:

_{Let G B}

(n;k) ## a bipartite

## graph with bipartite adjacency matrix has the form

## B

(n;k)## =

## 0

## B

## B

## B

## B

## B

## B

## B

## @

## a

1## a

2## : : :

## a

m## 0 : : : 0

## 1

## 0

_{F}

(n 1;2)
## ..

## .

## 0

## 1

## C

## C

## C

## C

## C

## C

## C

## A

## ;

## for some elements a

1## ; a

2## ; : : : ; a

m## in a ring R:Then the number of perfect matching

## of G B

(n;k)## is nth (k; )-number s

k## (n)

## with

## = (a

1## ; a

2## ; : : : ; a

m## )".

## bi-partite adjacency matrix has the form

## V

n## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 0

## 0

## 0

## 0

## 1

## 1

## 1

## 0

## 0

## ..

## .

## . .. ... ... ... ... ...

## ..

## .

## . .. ... ... ... 0

## ..

## .

## . .. ... ... 1

## 0

## 0

## 1

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## with the entries are

## v

ij## =

## (

## 1;

## if

## 1

## j

## i

## 1

## or i = 1;

## 0;

## otherwise

## ". Then they obtain that "the number of perfect matchings of G (V

n## )

## is

## P

ni=0

## F (i) =

## F (n + 2)

## 1;

## where F (n) is the nth Fibonacci number". They also consider "a

## bipartite graph G (W

n## )

## with bipartite adjacency matrix W

n## = V

n## + Y

n## ;

## where Y

n## de-notes the n

## n

## matrix with

## 1

## at the (1; 2)-entry, 1 at the (2; 4)-entry and zeros

## elsewhere". Clearly,

## W

n## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 0

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 0

## 0

## 0

## 1

## 1

## 1

## 0

## 0

## ..

## .

## . .. ... ... ... ... ...

## ..

## .

## . .. ... ... ... 0

## ..

## .

## . .. ... ... 1

## 0

## 0

## 1

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## Then for n

## 4;

## they obtain that "the number of perfect matchings of G (W

n## )

## is

## P

n 2i=0

## L (i) = L (n)

## 1;

## where L (n) is the nth Lucas number" (K¬l¬ç and Ta¸

## sc¬,

## 2008a).

## K¬l¬ç and Ta¸

## sc¬(2008b) consider "a bipartite graph G M

(n;k)## with bipartite

## adjacency matrix M

(n;k)## =

## F

(n;k)## +U

(n;k)## ;

## where F

(n;k)## is the matrix given by (3) and

## 0": Clearly,

## M

(n;k)## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 1

## 1

## 0

## 0

## 0

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## 0

## 0

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## 0

## ..

## .

## . .. ... ... ... ... ... ... ... ...

## ..

## .

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## ..

## .

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## ..

## .

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## 1

## 1

## 1

## 0

## 0

## 1

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## :

## (4)

## Then for n

## 3;

## they obtain that "the number of perfect matchings of G M

(n;k)## is

## the nth k-Lucas number, l

k## (n)". They also consider "a bipartite graph G T

(n;k)## with bipartite adjacency matrix T

(n;k)## =

## F

(n;k)## + V

(n;k)## for 2

## k < n;

## where F

(n;k)## is the matrix given by (3) and V

(n;k)## = (v

ij## )

## is the n

## n (0; 1)-matrix with v

1j## = 1

## if k + 1

## j

## n

## and otherwise 0": Clearly,

## T

(n;k)## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 0

## 0

## 0

## 0

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## ..

## .

## . .. ... ... ... ... ... ... ...

## ..

## .

## ..

## .

## 0

## 1

## 1

## 1

## 1

## . .. ...

## ..

## .

## 0

## 1

## 1

## 1

## 1

## 0

## ..

## .

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## 1

## 1

## 1

## 0

## 0

## 1

## 1

## 1

## 0

## 0

## 1

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## :

## Then they obtain that "the number of perfect matchings of G T

(n;k)## is the sums

## of k-Fibonacci numbers,

## P

n_{j=1}

## g

k## (j)

## for n

## 3" (K¬l¬ç and Ta¸

## sc¬, 2008b). This

## time, they consider "a bipartite graph G E

(n;k)## with bipartite adjacency matrix

## E

(n;k)## = M

(n;k)## + D

(n;k)## for 2

## k < n;

## where M

(n;k)## is the matrix given by (4) and

## 0": Clearly,

## E

(n;k)## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 0

## 0

## 0

## 0

## 0

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## 0

## 0

## ..

## .

## . .. ... ... ... ... ... ... ... ... ...

## ..

## .

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## ..

## .

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## ..

## .

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## 0

## 1

## 1

## 1

## 1

## 0

## 0

## 1

## 1

## 1

## 0

## 0

## 1

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## :

## Then they obtain that "the number of perfect matchings of G E

(n;k)## is the sums

## of k-Lucas numbers,

## P

n 1_{j=1}

## l

k## (j)

## for n

## 3" (K¬l¬ç and Ta¸

## sc¬, 2008b).

## K¬l¬ç and Stakhov (2009) consider "a bipartite graph G M

(n;p)## with

## bipar-tite adjacency matrix M

(n;p)## = (m

ij## )

## with m

i+1;i## = m

i;i## = m

i;i+p## = 1

## for a …xed

## integer p and otherwise 0": Clearly,

## M

(n;p)## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 0

## 0

## 1

## 0

## 0

## 1

## 1

## 0

## 0

## 1

## . .. ...

## 0

## 1

## 1

## 0

## 0

## 1

## 0

## 0

## 0

## 1

## 1

## 0

## 0

## 1

## ..

## .

## . .. ... ... ... ...

## 0

## ..

## .

## . .. ... ... ... ... ...

## ..

## .

## . .. ... ... ... 0

## 0

## 0

## 0

## 1

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## :

## (5)

## Then they obtain that "the number of perfect matchings of G M

(n;p)## is the (n + 1)st

## generalized Fibonacci p-number F

p## (n + 1)

## for n

## 3". They also consider a bipartite

## graph G T

(n;p)## with bipartite adjacency matrix has the form

## T

(n;p)## =

## 0

## B

## B

## B

## B

## B

## B

## B

## @

## 1 1 1

## 1 1

## 1

## 0

## M

(n 1;p)## ..

## .

## 0

## 1

## C

## C

## C

## C

## C

## C

## C

## A

## ;

## where M

(n 1;p)## is the matrix of order (n

## 1)

## given by (5)". Then they obtain

## that "the number of perfect matchings of G T

(n;p)## is the sums of the consecutive

## generalized Fibonacci p-numbers,

## P

n_{i=1}

## F

p## (i)

## for n

## 3" (K¬l¬ç and Stakhov, 2009).

## Akbulak and Ötele¸

## s (2013) give the following results for the number of

## per-fect matchings in bipartite graphs with upper Hessenberg adjacency matrix related

## to Fibonacci, Lucas and Padovan numbers.

## "Let G(U

n## = (u

ij## ))

## be a bipartite graph with bipartite adjacency matrix is

## the n

## n (0; 1)-upper Hessenberg matrix de…ned by

## U

n## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 0

## 1

## 0

## 1

## 0

## 1

## 1

## 1

## 0

## 1

## 0

## 1

## 0

## 1

## 0

## 1

## 1

## 0

## 1

## 0

## 1

## 0

## 1

## 0

## 0

## 1

## 1

## 0

## 1

## 0

## 1

## 0

## 1

## 0

## 0

## 1

## 1

## 0

## 1

## 0

## 1

## 0

## 0

## 0

## 1

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## (6)

## where

## u

ij## =

## 8

## >

## <

## >

## :

## 1;

## if j

## i =

## 1;

## 1;

## if i

## j

## and j

## i

## 0 (mod 2) ;

## 0;

## otherwise.

## Then the number of perfect matchings of G(U

n## )

## is the nth Fibonacci number F (n)

## for n

## 3" (Akbulak and Ötele¸

## s, 2013).

## "Let G(V

n## = (v

ij## ))

## be a bipartite graph with bipartite adjacency matrix is

## the n

## n (0; 1)-upper Hessenberg matrix de…ned by

## V

n## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1 1 1

## 1

## 1 0 1

## 0

## 1

## 0 1

## 0 0

## U

n 2## ..

## .

## ..

## .

## 0 0

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## ;

## where U

n 2## is the matrix of order (n

## 2)

## in 6. Then the number of perfect matchings

## of G(V

n## )

## is the (n

## 1)st Lucas number L (n

## 1)

## for n

## 3" (Akbulak and Ötele¸

## s,

## 2013).

## the n

## n (0; 1)-upper Hessenberg matrix de…ned by

## W

n## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 1

## 0

## 1

## 0

## 0

## 1

## 0

## 0

## 1

## 0

## 1

## 0

## 0

## 1

## 0

## 0

## 0

## 1

## 0

## 1

## 0

## 0

## 1

## 0

## 0

## 0

## 1

## 0

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## ;

## where

## w

ij## =

## 8

## >

## <

## >

## :

## 1;

## if i = 1 or j

## i =

## 1;

## 1;

## if i

## j

## and j

## i

## 1 (mod 3) ;

## 0;

## otherwise.

## Then the number of perfect matchings of G(W

n## )

## is the nth Padovan number P (n)

## for n

## 3" (Akbulak and Ötele¸

## s, 2013).

## Ötele¸

## s (2017) gives the following results for the number of perfect matchings

## in bipartite graphs related to the well-known integer sequences (Fibonacci, Lucas

## i.e.).

## "Let G (H

n## = (h

i;j## )) (n = 2t; t

## 2 N) be a bipartite graph with bipartite

## adjacency matrix is the n

## n (0; 1)-tridiagonal matrix has the form

## H

n## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 1

## 0

## 0

## 1

## 0

## 1

## . ..

## ..

## .

## 0

## 1

## 1

## 1

## . ..

## ..

## .

## ..

## .

## . .. 1

## 0

## 1

## . .. ...

## ..

## .

## . .. 1 ... 1 0

## ..

## .

## . .. ...

## 1

## 0

## 0

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## ;

## (7)

## where

## h

i;j## =

## 8

## >

## >

## <

## >

## >

## :

## 1;

_{if jj}

## i

_{j = 1;}

## 1;

## if i = j = 2m

## 1 (m

_{2 N) ;}

## 0;

## otherwise.

## Then the number of perfect matchings of G (H

n## )

## is 1" (Ötele¸

## s, 2017).

## "Let G (H

n## ) (n = 2t + 1; t

## 2 N) be a bipartite graph with bipartite

## adja-cency matrix H

n## given by (7). Then the number of perfect matchings of G (H

n## )

## is

n+1

2

## " (Ötele¸

## s, 2017).

## the n

## n (0; 1)-tridiagonal matrix has the form

## K

n## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 1

## 0

## 0

## 1

## 0

## 1

## . ..

## ..

## .

## 0

## 1

## 1

## 1

## . ..

## ..

## .

## ..

## .

## . .. 1

## 0

## 1

## . .. ...

## ..

## .

## . .. 1

## 1

## 1

## 0

## ..

## .

## . .. ... ... 1

## 0

## 0

## 1

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## ;

## where k

2;2## = k

4;4## = 0;

## all other terms on the main diagonal are 1; all terms on the

## sub-diagonal and superdiagonal are 1 and otherwise k

i;j## = 0. Then for n

## 3, the

## number of perfect matchings of G (K

n## )

## is the (n

## 3)rd Lucas number L (n

## 3)"

## (Ötele¸

## s, 2017).

## "Let G (B

n## = (b

i;j## ))

## be a bipartite graph with bipartite adjacency matrix is

## the n

## n (0; 1)-anti-tridiagonal matrix has the form

## B

n## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 0

## 0

## 1

## 1

## ..

## .

## . .

## .

## 1

## 1

## 1

## ..

## .

## . .

## .

## . .

## .

## . .

## .

## 1

## 0

## 0

## . .

## .

## . .

## .

## . .

## .

## . .

## .

## ..

## .

## 1

## 1

## . .

## .

## . .

## .

## ..

## .

## 1

## 1

## 0

## 0

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## ;

## where

## b

i;j## =

## (

## 1;

_{if ji + j mod (n + 1)j}

## 1;

## 0;

## otherwise.

## Then for n

## 2, the number of perfect matchings of G (B

n## )

## is the (n + 1)st Fibonacci

## number F (n + 1)" (Ötele¸

## s, 2017).

## "Let G (D

n## = (d

i;j## ))

## be a bipartite graph with bipartite adjacency matrix

## has the form

## D

n## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 0

## 0

## 1

## 0

## 1

## ..

## .

## 0

## 1

## 1

## 1

## ..

## .

## . .

## .

## . .

## .

## . .

## .

## 1

## 0

## 0

## . .

## .

## . .

## .

## . .

## .

## . .

## .

## ..

## .

## 1

## 1

## 1

## . .

## .

## ..

## .

## 1

## 1

## 0

## 0

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## where

## d

i;j## =

## 8

## >

## >

## <

## >

## >

## :

## 1;

## if i = 1 and j = n

## 2; j = n;

## 1;

_{if ji + j mod (n + 1)j}

_{2 i n}

## 1;

## 0;

## otherwise.

## Then for n

## 3, the number of perfect matchings of G (D

n## )

## is the (n

## 1)st Lucas

## number L (n

## 1)" (Ötele¸

## s, 2017).

## "Let G (U

n## = (u

i;j## ))

## be a bipartite graph with bipartite adjacency matrix

## has the form

## U

n## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 1

## 1

## 1

## 1

## 0

## 0

## 1

## 1

## 1

## ..

## .

## . .

## .

## 1

## 1

## 1

## 0

## ..

## .

## . .

## .

## . .

## .

## . .

## .

## 1

## . .

## .

## 0

## 1

## . .

## .

## . .

## .

## . .

## .

## ..

## .

## 1

## 1

## 1

## . .

## .

## 0

## 1

## 1

## 0

## 0

## 0

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## where

## u

i;j## =

## 8

## >

## >

## <

## >

## >

## :

## 1;

## if i = 1 and 1

## j

## n;

## 1;

_{if ji + j mod (n + 1)j}

_{2 i n}

## 1;

## 0;

## otherwise.

## Then for n

## 2, the number of perfect matchings of G (U

n## )

## is F (n + 2)

## 1, where

## F (n)

## is the nth Fibonacci number" (Ötele¸

## s, 2017).

## "Let G (V

n## = (v

i;j## ))

## be a bipartite graph with bipartite adjacency matrix

## has the form

## V

n## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 1

## 1

## 1

## 0

## 1

## 0

## 0

## 1

## 1

## 1

## 1

## ..

## .

## . .

## .

## 0

## 1

## 1

## 1

## 0

## ..

## .

## . .

## .

## . .

## .

## . .

## .

## 1

## 1

## 0

## 0

## 0

## . .

## .

## . .

## .

## . .

## .

## 1

## . .

## .

## . .

## .

## ..

## .

## 0

## 1

## 1

## . .

## .

## . .

## .

## . .

## .

## ..

## .

## 1

## 1

## 1

## 0

## 0

## 0

## 1

## 1

## 0

## 0

## 0

## 0

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## where

## v

i;j## =

## 8

## >

## >

## >

## >

## >

## <

## >

## >

## >

## >

## >

## :

## 1;

## if i = 1 and 1

## j

## n

## 2; j = n;

## 1;

## if i = 2 and j = n

## 3;

## 1;

_{if ji + j mod (n + 1)j}

_{2 i n}

## 1;

## 0;

## otherwise.

## is the nth Lucas number" (Ötele¸

## s, 2017).

## "Let G (P

n## = (p

ij## ))

## be a bipartite graph with bipartite adjacency matrix is

## the n

## n (0; 1)-pentadiagonal matrix has the form

## P

n## =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## 1

## 0

## 1

## 0

## 0

## 0

## 1

## 0

## 1

## . ..

## ..

## .

## 1

## 0

## 1

## . .. ... ... ...

## 0

## 1

## . .. ... ... 1 0

## ..

## .

## . .. ... ... ... 0 1

## ..

## .

## . .. 1

## 0

## 1

## 0

## 0

## 0

## 1

## 0

## 1

## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## :

## Then the number of perfect matchings of G (P

n## )

## is obtained as

## per (P

n## ) =

## 8

## <

## :

## F

n+1_{2}

## F

n+3_{2}

## ;

## if n is odd,

## F

n_{2}

## + 1

2## ;

## if n is even,

## 3. PRELIMINARIES

## In this chapter, we give the fundamental de…nitions, structures and theorems

## which are necessary to better understand the topics contained within this text.

## 3.1. Special Types of Matrices

## 3.1.1. Diagonal matrices

## "A matrix D = [d

ij## ]

## 2 M

n;m## (F )

## is diagonal if d

ij## = 0

## whenever j 6= i" (Horn

## and Johnson, 1990).

## 3.1.2. Triangular matrices

## "A matrix T = [t

ij## ]

## 2 M

n;m## (F )

## is upper triangular if t

ij## = 0

## whenever

## i > j. If t

ij## = 0

## whenever i

## j

## , then T is said to be strictly upper triangular.

## Analogously, T is lower triangular (or strictly lower triangular) if its transpose is

## upper triangular (or strictly upper triangular). A triangular matrix is either lower

## or upper triangular; a strictly triangular matrix is either strictly upper triangular

## or strictly lower triangular" (Horn and Johnson, 1990).

## 3.1.3. Permutation matrices

## "A square matrix P is a permutation matrix if exactly one entry in each

## row and column is equal to 1 and all other entries are 0. Multiplication by such

## matrices e¤ects a permutation of the rows or columns of the matrix multiplied. For

## example,

_{0}

## B

## @

## 0 1 0

## 1 0 0

## 0 0 1

## 1

## C

## A

## 0

## B

## @

## 1

## 2

## 3

## 1

## C

## A =

## 0

## B

## @

## 2

## 1

## 3

## 1

## C

## A

## illustrates how a permutation matrix produces a permutation of the rows (entries)

## of a vector: it sends the …rst entry to the second position, sends the second entry

## to the …rst position, and leaves the third entry in the third position" (Horn and

## Johnson, 1990).

## 3.1.4. Circulant matrices

## "A matrix A 2 M

n## (F )

## of the form

## A =

## 0

## B

## B

## B

## B

## B

## B

## B

## @

## a

1## a

2## a

n## a

n## a

1## a

2## a

n 1## a

n 1## a

n## a

1## a

n 2## ..

## .

## ..

## .

## . .. ...

## ..

## .

## a

2## a

3## a

n## a

1## 1

## C

## C

## C

## C

## C

## C

## C

## A

## is a circulant matrix. Each row is the previous row cycled forward one step; the

## en-tries in each row are a cyclic permutation of those in the …rst. The n n permutation

## matrix

## C =

## 0

## B

## B

## B

## B

## B

## B

## B

## @

## 0 1

## 0

## 0

## ..

## .

## 0

## 1

## ..

## .

## . .. ... 0

## 0

## 1

## 1 0

## 0

## 1

## C

## C

## C

## C

## C

## C

## C

## A

## =

## 0

## I

n 1## 1 0

1;n 1## !

## is the basic circulant permutation matrix. A matrix A 2 M

n## (F )

## can be written in

## the form

## A =

n 1## X

k=0## a

k+1## C

nk## (a polynomial in the matrix C

n## ) if and only if it is a circulant. We have C

n0## = I =

## C

nn

## ;

## and the coe¢ cients a

1## ; : : : ; a

n## are the entries of the …rst row of A" (Horn and

## Johnson, 1990).

## 3.1.5. Toeplitz matrices

## "A matrix A = a

ij## 2 M

n+1## (F )

## of the form

## A =

## 0

## B

## B

## B

## B

## B

## B

## B

## B

## B

## B

## @

## a

0## a

1## a

2## a

n## a

1## a

0## a

1## a

2## a

n 1## a

2## a

1## a

0## a

1## a

n 2## ..

## .

## ..

## .

## . .. ... ...

## ..

## .

## ..

## .

## ..

## .

## . .. ...

## a

1## a

n## a

n+1## a

1## a

0## 1

## C

## C

## C

## C

## C

## C

## C

## C

## C

## C

## A

## is a Toeplitz matrix.

## The entry a

ij## is equal to a

j i## for some given sequence

## the diagonals parallel to the main diagonal. The Toeplitz matrices

## B =

## 0

## B

## B

## B

## B

## B

## @

## 0 1

## 0

## 0

## . ..

## . .. 1

## 0

## 0

## 1

## C

## C

## C

## C

## C

## A

## and

## F =

## 0

## B

## B

## B

## B

## @

## 0

## 0

## 1

## 0

## . .. ...

## 0

## 1

## 0

## 1

## C

## C

## C

## C

## A

## are called the backward shift and forward shift because of their e¤ect on the elements

## of the standard basis fe

1## ; : : : ; e

n+1## g. Moreover, F = B

T## and B = F

T## . A matrix

## A

_{2 M}

n+1 ## can be written in the form

## A =

n## X

k=1## a

k## F

k## +

n## X

k=0## a

k## B

k## if and only if it is a Toeplitz matrix. Toeplitz matrices arise naturally in problems

## involving trigonometric moments" (Horn and Johnson, 1990).

## 3.1.6. Hankel matrices

## "A matrix A 2 M

n+1## (F )

## of the form

## A =

## 0

## B

## B

## B

## B

## B

## B

## B

## @

## a

0## a

1## a

2## a

n## a

1## a

2## . .

## .

## a

n+1## a

2## . .

## .

## ..

## .

## ..

## .

## a

n## a

2n 1## a

n## a

n+1## a

2n 1## a

2n## 1

## C

## C

## C

## C

## C

## C

## C

## A

## is a Hankel matrix . Each entry a

ij## is equal to a

i+j 2## for some given sequence

## a

0## ; a

1## ; a

2## ; : : : ; a

2n 1## ; a

2n## :

## The entries of A are constant along the diagonals

## perpen-dicular to the main diagonal. Hankel matrices arise naturally in problems involving

## power moments" (Horn and Johnson, 1990).

## 3.1.7. Hessenberg matrices

## be an upper Hessenberg matrix if a

ij## = 0

## for all i > j + 1 :

## A =

## 0

## B

## B

## B

## B

## B

## B

## B

## @

## a

11## F

## a

21## a

22## a

32## . ..

## . ..

## . ..

## 0

## a

n;n 1## a

nn## 1

## C

## C

## C

## C

## C

## C

## C

## A

## An upper Hessenberg matrix A is said to be unreduced if all its sub-diagonal entries

## are nonzero, that is, if a

i+1## ; i = 0

## for all i = 1; : : : ; n

## 1;

## the rank of such a matrix

## is at least n

## 1

## since its …rst n

## 1

## columns are independent" (Horn and Johnson,

## 1990).

## 3.1.8. Tridiagonal matrices

## "A matrix A = [a

ij## ]

## 2 M

n## (F )

## that is both upper and lower Hessenberg is

## called tridiagonal, that is, A is tridiagonal if a

ij## = 0

## whenever ji

## j

## j > 1:

## A =

## 0

## B

## B

## B

## B

## B

## @

## a

1## b

1## 0

## c

1## a

2## . ..

## . .. ... b

n 1## 0

## c

n 1## a

n## 1

## C

## C

## C

## C

## C

## A

## " (Horn and Johnson, 1990).

## 3.2. Graph Theory

## De…nition 1

_{"A graph G consists of a set of objects V = fv}

1## ; v

2## ; v

3## ; : : :

## g called

## ver-tices (also called points or nodes) and other set E = fe

1## ; e

2## ; e

3## ; : : :

## g whose elements

## are called edges (also called lines or arcs)" (Vasudev, 2006).

## De…nition 2

## "The set V (G) is called the vertex set of G and E(G) is the edge set.

## Usually the graph is denoted as G = (V; E)" (Vasudev, 2006).

## Let G be a graph and fu; vg an edge of G. Since fu; vg is 2-element set, we may

## write fv; ug instead of fu; vg. It is often more convenient to represent this edge by

## uv

## or vu: If e = uv is an edge of a graph G, then we say that u and v are adjacent in

## G

## and that e joins u and v. (We may also say that each that of u and v is adjacent

## to or with the other).

## and

## E(G) =

_{fuv; uw; wx; xy; xzg:}

## Now we have the graph in the Figure 3.1 by considering these sets.

Figure 3.1. A graph with

## 6

-vertices and## 5

-edges## Every graph has a diagram associated with it. The vertex u and an edge e

## are incident with each other as are v and e. If two distinct edges say e and f are

## incident with a common vertex, then they are adjacent edges.

## Example 4

## In Figure 3.2 the vertices a and b are adjacent but a and c are not.

## The edges x and y are adjacent but x and z are not. Although the edges x and z

## intersect in the diagram, their intersection is not a vertex of the graph.

Figure 3.2. A graph with

## 5

-vertices and## 7

-edges## De…nition 5

## "A graph with p-vertices and q-edges is called a (p; q) graph. The

## (1; 0) graph is called trivial graph" (Vasudev, 2006).

## Example 6

_{Let V = f1; 2; 3; 4g and E = ff1; 2g; f1; 3g; f3; 2g; f4; 4gg:Then G(V; E)}

## is a graph.

## Example 7

_{Let V = f1; 2; 3; 4g and E = ff1; 5g; f2; 3gg:Then G(V; E) is not a}

Figure 3.3. A graph with

## 5

-vertices and## 8

-edges is called a## (5; 8)

graph## De…nition 8

## "A directed graph or digraph G consists of a set V of vertices and a

## set E of edges such that e 2 E is associated with an ordered pair of vertices. In other

## words, if each edge of the graph G has a direction then the graph is called directed

## graph" (Vasudev, 2006).

## In the diagram of directed graph, each edge e = (u; v) is represented by an

## arrow or directed curve from initial point u of e to the terminal point v. Figure 3.4

## is an example of a directed graph.

Figure 3.4. Directed graph

## Suppose e = (u; v) is a directed edge in a digraph, then

## (i) u is called the initial vertex of e and v is the terminal vertex of e

## (ii) e is said to be incident from u and to be incident to v.

## (iii) u is adjacent to v, and v is adjacent from u.

## De…nition 9

## "An un-directed graph G consists of set V of vertices and a set E of

## edges such that each edge e 2 E is associated with an unordered pair of vertices. In

## other words, if each edge of the graph G has no direction then the graph is called

## We

## can refer to an edge joining the vertex pair i and j as either (i; j) or

## (j;

## i): Figure 3.5 is an example of an undirected graph.

Figure 3.5. Undirected graph

## De…nition 10

## "An edge of a graph that joins a node to itself is called loop or self

## loop" (Vasudev, 2006).

## For example, a loop is an edge (v

i## ; v

j## )

## where v

i## = v

j## .

## De…nition 11

## "A multi-graph is a graph which is permitted to have multiple edges

## (also called parallel edges) that is, edges that have the same end nodes" (Vasudev,

## 2006).

## Two edges (v

i## ; v

j## )

## and (v

f## ; v

r## )

## are parallel edges if v

i## = v

f## and v

j## = v

r## .

Figure 3.6. Directed multiple graph Figure 3.7. Un-directed multiple graph

## In Figure 3.6, there are two parallel edges associated with v

2## and v

3## . In Figure 3.7,

## there are two parallel edges joining nodes v

1## and v

2## and v

2## and v

3## (Vasudev, 2006).

## De…nition 12

## "A graph in which loops and multiple edges are allowed, is called a

## pseudo graph.

Figure 3.8. Un-directed Pseudo graph Figure 3.9. Directed Pseudo graph

## ". (Vasudev, 2006).

## De…nition 13

## "A graph which has neither loops nor multiple edges. i.e., where

## each edge connects two distinct vertices and no two edges connect the same pair of

## vertices is called a simple graph" (Vasudev, 2006).

## Figure 3.4 and 3.5 represents simple un-directed and directed graph because

## the graphs do not contain loops and the edges are all distinct.

## De…nition 14

## "A graph with …nite number of vertices as well as a …nite number of

## edges is called a …nite graph. Otherwise, it is an in…nite graph" (Vasudev, 2006).

## De…nition 15

## "The number of edges incident on a vertex v

i## with self-loops counted

## twice (is called the degree of a vertex v

i## and is denoted by deg

G## (v

i## ) or degv

i## or d(v

i## )"

## (Vasudev, 2006).

## The degrees of vertices in the graph G and H are shown in Figure 3.10 and

## 3.11.

## and

Figure 3.11. A graph with

## 5

-vertices and## 8

-edges## In G as shown in Figure 3.10, deg

G## (v

2## ) = 2 = deg

G## (v

4## ) = deg

G## (v

1## ), deg

G## (v

3## ) = 3

## and

## deg

_{G}

## (v

5## ) = 1

## In H as shown in Figure 3.11 deg

H## (v

2## ) = 5; deg

H## (v

4## ) = 3; deg

H## (v

3## ) =

## 5; deg

_{H}